Intrinsic topological superconductivity remains one of the central open questions in condensed-matter physics. The three-dimensional Weyl semimetal PtBi₂ has recently been shown by angle-resolved photoemission spectroscopy to host a superconducting gap that opens exclusively on its Fermi-arc surface states and exhibits a nodal structure, making it a promising platform for exploring unconventional and potentially topological superconductivity. However, despite these momentum-resolved insights, the real-space spectroscopic signatures of unconventional superconductivity in PtBi₂ remain unclear. Here, using scanning tunneling microscopy and spectroscopy, we reveal a sizable and spatially homogeneous gap (Δ > 10 meV) on atomically pristine terraces of PtBi₂. Within this gap, we detect surface-extended in-gap states exhibiting pronounced atomic-scale spatial modulation. Analysis of the tunneling spectra indicates that these states are most consistently interpreted as Andreev bound states within a superconducting gap. Further phenomenological modeling shows that their characteristic spectral features are naturally captured by a chiral pairing scenario. Our results establish a direct real-space connection between the superconducting gap and surface-extended quasiparticle states in PtBi₂, and provide new insight into the nature of its unconventional surface superconductivity. These findings highlight PtBi₂ as a versatile platform for exploring exotic quasiparticle excitations and potential Majorana physics.